Stabilizing solutions to output feedback pole placement problem with parameter drift and automated alerting of system parameter changes

ABSTRACT

Output feedback pole placement problems with parameter drift are solved with stabilizing solutions. Changes in system parameters trigger alerts in an automated manner. A representative method includes determining a set of solutions for an output feed pole placement problem, based on parameters of a physical system. The solutions are stable and well-conditioned for monitoring changes to the parameters of the physical system. The physical system is adjusted, or controlled, based on the solutions determined. Updated parameters of the physical system are acquired. A set of updated solutions for the output feedback pole placement problem are determined based on the updated parameters. The physical system is then adjusted, or controlled, based on the updated solutions determined. A system manager may also be notified of the updated parameters and/or the updated solutions. Furthermore, changes within the system may be monitored, and/or potentially critical changes within the system may be detected.

FIELD OF THE INVENTION

The present invention relates generally to solutions to the outputfeedback pole placement problem, and more particularly to such solutionsthat accommodate parameter drift, and automatically alert when changesin system parameters occur.

BACKGROUND OF THE INVENTION

The output feedback pole placement problem occurs in physical systems,such as mechanical, scientific, engineering, geometrical, and biologicalsystems. A representative example of such a system 100 is depicted inFIG. 1. In particular, there are mechanical linkages 104A, 104B, and104C, collectively referred to as the mechanical linkages 104. Thelinkages 104 are connected to each other and to a surface 102, such as awall, at points 106A, 106B, and 106C, collectively referred to as thejoints 106, as depicted in FIG. 1. The linkages 104 may rotate aroundthe joints 106 to which they are connected.

The mechanical linkages 104 are driven by torques u_(i) applied to thejoints 106 with measured angular displacements y_(i). It may be desiredto understand and control the behavior of this system 100. Settingv_(i):=ith angular velocity, the system 100 thus evolves according tothe linearized Newton equations:

$\begin{matrix}{\frac{\mathbb{d}v_{i}}{\mathbb{d}t} = {I_{i}u_{i}}} & (1) \\{\frac{\mathbb{d}y_{i}}{\mathbb{d}t} = v_{i}} & (2)\end{matrix}$

More generally, a physical system can be considered as having m inputsand p outputs, which are modeled as vectors u in R^(m) and y in R^(p).If this system is linear, or is at near equilibrium, then there are ninternal states x, which are considered as a vectors in R^(n), such thatthe system is governed by first order linear evolution equations:

$\begin{matrix}{\frac{\mathbb{d}x}{\mathbb{d}t} = {{Ax} + {Bu}}} & (3) \\{y = {Cx}} & (4)\end{matrix}$FIG. 2 shows a schematic representation of such a physical system 200.The Fourier transform of equation (3) gives sx=Ax+Bu. If this is solvedfor x, and substituted in equation (4), the following result isobtained:y=C(sI−A)⁻¹ Bu  (5)The multiplier C(sI−A)⁻¹B is called the transfer function of the system.This p by m matrix of rational functions determines the response of themeasured quantities y in terms of the inputs u, in the frequency domain.

Now, it is supposed that the system 200 is wished to be controlled witha constant linear output feedback u=Fy. Such a corresponding physicalsystem 300 is depicted in FIG. 3. The behavior of the closed system

$\begin{matrix}{\frac{\mathbb{d}x}{\mathbb{d}t} = {\left( {A + {BFC}} \right)x}} & (6)\end{matrix}$is determined by the roots of the characteristic polynomialf(s)=det(sI _(n) −A−BFC)  (7)Thus, the forward problem is, given a physical system, represented asmatrices A, B, C, and a feedback law F, then the system evolvesaccording to the behavior encoded in its characteristic polynomial f(s).The inverse problem is the pole placement problem. That is, given alinear system represented by matrices A, B, C, and a desired behaviorf(s), which feedback laws F satisfy f(s)=det(sI_(n)−A−BFC)?

The generalized static output feedback pole place problem can thereforebe described as follows. Given real matrices AεR^(n×n), BεR^(n×m),CεR^(p×n) and closed subsets C₁, C₂, . . . , C_(n)⊂C, find KεR^(m×p)such thatλ(A+BKC)εC _(i) for i=1, 2, . . . , n  (8)Here, λ_(i)(A+BKC) denotes the ith eigenvalues of A+BKC. Sucheigenvalues can be considered the parameters of the physical systembeing analyzed and/or controlled.

There are various special cases of the generalized static outputfeedback pole placement problem. For instance, the classical poleplacement problem can be written asC_(i)={c_(i)}, c_(i)εC  (9)The regions C_(i) are discrete points. Stabilization-type problems forcontinuous time and discrete time physical systems can be written asC ₁ =C ₂ = . . . =C _(n) ={zεC|Re(z)≦−α}  (10)andC ₁ =C ₂ = . . . =C _(n) ={zεC||z|≦α}  (11)respectively. The relaxed classical pole placement problem is written asC ₁ =C ₂ = . . . =C _(n) ={zεC||z−c _(i) |≦r _(i)}  (12)

Furthermore, a final special case of the generalized static outputfeedback pole placement problem includes hybrid pole placement problems.These are problems that specify a pair of poles must be placed at apoint and its complex conjugate, while all other belong within a closedregion C. Thus,C₁{c}, C₂={ c}, C₃=C₄= . . . C_(n)=C  (13)In variations of the hybrid pole placement problem, more than n pairs ofpoints must be placed at a set of n points and their correspondingcomplex conjugates, while all others belong to a closed region C.

Solutions to these and other output feedback pole placement problemstypically assume that the system parameters are constant. However, inmany if not most real-world physical systems, the parameters are notperfectly constant. For example, everyday wear-and-tear on mechanicalsystem parts can cause parameters to slowly drift from the originalinitial values.

The reference Kaiyang Yang and Robert Orsi, “Pole Placement via OutputFeedback: A Methodology Based on Projections,” in Proceedings of the16th IFAC World Congress, Prague, Czech Republic, 2005, hereinafterreferred to as [Yang and Orsi], discloses an algorithm for solving thegeneralized static output feedback pole placement problem describedabove in relation to equation (8). The algorithm of [Yang and Orsi],however, assumes that system parameters are perfectly constant orperfectly static. As a result, the algorithm cannot accommodate manyreal-world physical systems, in which such parameters are not perfectlyconstant.

Furthermore, the reference Kaiyang Yang, Robert Orsi, and John B. Moore,“A Projective Algorithm for Static Output Feedback Stabilization,” inProceedings of the 2nd IFAC Symposium on System, Structure and Control,Oaxaca, Mexico, 2004, hereinafter referred to as [Yang, Orsi, andMoore], a projective algorithm is proposed for the following staticoutput feedback problem. Given a linear time invariant (LTI) system

$\begin{matrix}{\frac{\mathbb{d}x}{\mathbb{d}t} = {{Ax} + {Bu}}} & (14) \\{y = {Cu}} & (15)\end{matrix}$where the vectors xεR^(n), uεR^(m), yεR^(p), and the matrices AεR^(n×n),BεR^(n×m), CεR^(p×n), find a static output feedback control lawu=Ky  (16)where KεR^(m×p) is a constant matrix such that the eigenvalues k_(i) ofthe resulting n-by-n close-loop system matrix A+BKC have non-positivereal parts.

The projective algorithm of [Yang, Orsi, and Moore], like the algorithmof [Yang and Orsi] and other solutions to static output feedbackproblems, assumes that system parameters are perfectly constant orperfectly static. Because system parameters are typically not constant,however, the solutions provided by the projective algorithm of [Yang,Orsi, and Moore], [Yang and Orsi], and other solutions are not ideal.Furthermore, such solutions are often highly sensitive to small changesin the system parameters. Thus, even minor amounts of parameter driftcan cause these solutions to no longer be appropriate and useful.

In addition, because the controllers in such physical systems often relyon solutions to the output feedback pole placement problem that aredesigned using initial system parameter values, they are not equipped tomonitor drafts in system parameters over long periods of time.Furthermore, such controllers are not equipped to update the feedbackcontrol system to reflect these changes in the system parameters, noralert the system manager when critical changes in the system parametersoccur. For example, the controller in question may no longer be capableof accommodating the changes that have occurred over time. The netresult is the system damage, or safety problems, can result.

For these and other reasons, there is a need for the present invention.

SUMMARY OF THE INVENTION

The present invention relates to a method to increase the stability ofsolutions to the generalized output feedback pole placement problem.Furthermore, the method can monitor changes in system parameters, updatesolutions to the generalized output feedback pole placement problem toreflect changes in system problems, and detect possibly critical changesin the system parameters so that the system manager can be notified in atimely manner.

In general, the present invention increases the stability of solutionsto the almost generalized output feedback pole placement problem byusing more than one solution when system parameters are almost, but notperfectly, constant. In one embodiment, enhancements to the solution tothe generalized static output feedback pole placement problem proposedin [Yang and Orsi] are utilized. Furthermore, randomized sampling isutilized to select initial points for the iteration in the projectivealgorithm of [Yang, Orsi, and Moore], followed by testing to selectsolutions and iteration paths with attractive properties, such asstability to perturbations, noise and numerical round-off error, andthat result in well-conditioned solutions. The computational resultsfrom previous time steps are used to reduce monitoring costs.

The enhancements to the solution proposed in [Yang and Orsi] are asfollows. First, multiple solutions are determined, and just thosesolutions that are relatively stable to small perturbations areselected. Second, these solutions to the generalized static outputfeedback pole placement problem are monitored. Third, a system alertingmechanism is provided such that when one of the solutions can no longerbe updated, a notification is provided to the system manager. Fourth,such solutions can be automatically replaced by a replacement solution,such as with the closest approximate solution, or the number ofsolutions can be reduced.

The present invention can thus be used be used to monitor changes insystem parameters, update the system controller to reflect changes insystem parameters over time, and detect possibly critical changes in thesystem parameters so that a system manager can be notified in a timelymanner. As has been noted, the present invention can be initialized sothat when one of the solution points reaches an alerting point,indicative, for instance, of a sudden large change, the system managercan be notified. The system may then subsequently automaticallydetermine and replace the solution with a new solution, replace thesolution with the closest approximate solution, or reduce the number ofsolutions used for monitoring system driver. The user may select whichof these actions will occur, either before the system begins, orinteractively.

A method of the invention thus determines a set of solutions for anoutput feedback pole placement problem, based on parameters of aphysical system. The solutions are stable and well-conditioned formonitoring changes to the parameters of the physical system. Thephysical system is adjusted based on the solutions determined, such asby using a controller of the system. Updated parameters of the physicalsystem are acquired, and a set of updated solutions for the outputfeedback pole placement problem are determined based on these updatedparameters. The physical system is finally adjusted based on the updatedsolutions determined.

Still other aspects, embodiments, and advantages of the presentinvention will become apparent by reading the detailed description thatfollows, and by referring to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The drawings referenced herein form a part of the specification.Features shown in the drawing are meant as illustrative of only someembodiments of the invention, and not of all embodiments of theinvention, unless otherwise explicitly indicated, and implications tothe contrary are otherwise not to be made.

FIG. 1 is a diagram of an example representative physical systemdemonstrating a pole placement problem.

FIG. 2 is a diagram of a schematic representation of the physical systemof FIG. 1.

FIG. 3 is a diagram of a schematic representation of a feedback-controlphysical system, in conjunction with which and in relation toembodiments of the invention may be practiced.

FIG. 4 is a flowchart of a method for updating solutions and monitoringdrifts in the output feedback pole placement problem, according to anembodiment of the invention.

FIG. 5 is a flowchart of a method for determining a set of solutions forthe output feedback pole placement problem that can be used inconjunction with the method of FIG. 4, according to an embodiment of theinvention.

FIG. 6 is a flowchart of a method for updating a solution for the outputfeedback pole placement problem that can be used in conjunction with themethod of FIG. 4, according to an embodiment of the invention.

FIG. 7 is a diagram of a chart that can be used to monitor the physicalsystem and for notifying or alerting the system manager when possiblycritical changes in system parameters have occurred, according to anembodiment of the invention.

DETAILED DESCRIPTION OF THE DRAWINGS

In the following detailed description of exemplary embodiments of theinvention, reference is made to the accompanying drawings that form apart hereof, and in which is shown by way of illustration specificexemplary embodiments in which the invention may be practiced. Theseembodiments are described in sufficient detail to enable those skilledin the art to practice the invention. Other embodiments may be utilized,and logical, mechanical, and other changes may be made without departingfrom the spirit or scope of the present invention. The followingdetailed description is, therefore, not to be taken in a limiting sense,and the scope of the present invention is defined only by the appendedclaims.

FIG. 3 shows a schematic representation of a feedback-control physicalsystem 300, in conjunction with which and in relation to embodiments ofthe invention may be practiced. The physical system 300 includes amonitored component 302, representing the actual components of thephysical system being monitored and that have physical parameters thatcan be adjusted. The physical system 300 also includes a monitoring orcontroller component 304. The component 304 is specifically a feedbackcomponent, that based on the results of the component 302, adjusts theparameters of the component 302 in a feedback manner.

The behavior of this closed system 300, as has been described, is

$\begin{matrix}{\frac{\mathbb{d}x}{\mathbb{d}t} = {\left( {A + {BFC}} \right)x}} & (17)\end{matrix}$and which is determined by the roots of the characteristic polynomialf(s)=det(sI _(n) −A−BFC)  (18)The pole placement problem for the system 300 is thus, given a linearsystem represented by matrices A, B, C, and a desired behavior f(s),what are the feedback laws that satisfy f(s)=det(sI_(n)−A−BFC),especially where the system parameters represented by the matrices A, B,C are not constant, and can change over time.

FIG. 4 shows a method 400 for updating solutions and monitoring driftsin the output feedback pole placement problem, according to anembodiment of the invention. There are eigenvalues k_(i) (A+B K C) ofthe n-by-n closed loop system matrix (A+B K C) that are specified. Themethod 400, like other methods of embodiments of the invention, can beimplemented at least in part as one or more computer programs orcomputer program parts, stored on a computer-readable medium of anarticle of manufacture. The computer-readable medium may, for instance,be a recordable data storage medium or a modulated carrier signal.

First, the method 400 determines a set of solutions for the outputfeedback pole placement problem based on parameters of a physical system(402). The output feedback pole placement problem may specifically bethe classical, static-output feedback pole placement problem that hasbeen described. FIG. 5 shows a method 500 for determining this set ofsolutions in 402, according to an embodiment of the invention.

First, at an initial time t=t₀, a starting point is selected fordetermining a solution based on the parameters of the physical system(502). The solution is then determined based on the starting point, withthe assumption that the parameters of the system are constant, using theprojective approach of [Yang and Orsi] and of [Yang, Orsi and Moore](504). A number of values are stored for the solution, including theeigenvalues of the solution matrix, the final few matrices of thesolution path of iterations that led to the solution, and theeigenvalues of each final matrix of the solution path (506). Thisprocess is repeated several times (508), with distinct, or differentstarting points for each solution. Therefore, as many distinct solutionsto the classical output feedback pole placement are obtained as desired.The starting points may be randomly selected, or pseudo-randomlyselected—for instance, as those starting points that are evenly ornon-evenly spaced on a rectangular coordinate grid. Furthermore, thenumber of solutions that are determined can vary.

In one embodiment, a random matrix with the same eigenvalues k_(i) as(A+B K C) is determined as follows. The given eigenvalues are placedalong the diagonal of an upper triangular matrix, and randomly generatedpositive real numbers for the non-zero entries are placed above thediagonal. The upper triangular matrix is then multiplied by a randomlygenerated unitary matrix on the right-hand side, and its inverse on theleft-hand side. This random matrix can then be used as the startingpoint to determine a solution with the projective approach of [Yang,Orsi, and Moore]. Other mechanisms for generating a random startingmatrix that has a good chance of converging may also be used. Thisprocess is repeated a number of times, starting with distinct, ordifferent, randomly generated initial matrices to obtain as manydistinct solutions to the classical static output feedback pole problemas desired.

Furthermore, it is noted that the projective approach of [Yang and Orsi]and of [Yang, Orsi, and Moore] is based on computing a sequence ofmatrices, and their Schur decompositions, that converge to a solution.(In general, a Schur decomposition of the pair (A, B) is, if A and B arecomplex, A=QSZH and B=QTZH, where Q and Z are unitary and S and T areupper triangular.) The sequence of matrices can in principle be verylong. However, empirical observations in numerical implementationstudies of real-world problems suggest that the sequence normallyconsists of five to ten matrices, and the size of the matrices as atlargest about fifty-by-fifty.

Sometimes a random starting point does not converge, or converges veryslowly. In such cases, the iteration may be aborted, and anotherstarting point selected, to begin a new iterative process to determine asolution.

Finally, it is noted that the computational cost of performing 502, 504,and 506 is small for small number of matrices that appear in outputfeedback pole placement problems. More specifically, the Schurdecomposition of a real matrix includes two steps. First, the realmatrix is converted to a matrix that is upper triagonal plus one lowerdiagonal band, using Householder transformations. The computational costis approximately

$\frac{14\; n^{3}}{3}$flops, where approximately

$\frac{10n^{3}}{3}$flops are required for the converted matrix and approximately

$\frac{4n^{3}}{3}$flops are required for the explicit computation of the unitarytransformation. Second, the real matrix is converted to upper triangularform using Householder reflections in a process known within the art aschasing or zero chasing. The computation cost is O(n²) flops.

Still referring to FIG. 5, the next step or act is to verify thestability of each of the solutions that was determined (510). That is,the stability of the matrices within the solution path during theiterative computations and that converge to a solution is verified. Inone embodiment, this is accomplished as follows. First, the conditionnumber for each solution matrix is determined (512). The conditionnumbers indicate the stability of the matrices under perturbations. Thecondition number in the matrix L2-norm is the ratio of the smallest andlargest eigenvalues. Next, the distances of the smallest few eigenvaluesof each of the solution matrices from the origin are determined (514).These distances indicate whether a given matrix has a likelihood ofchanging in the near future.

Thus, the set of solutions (and their matrices) are selected as thesolutions that are well-conditioned and that have distances of thesmallest eigenvalues of which being relatively far from the origin(516). Well-conditioned solutions are solutions having solution matricesthat have relatively large condition numbers, indicating that thesesolutions are stable under perturbations, and thus in response tochanges in the system parameters. That is, such solutions are morestable, and less sensitive to noise and numerical round-off errors.Solutions that have eigenvalues with distances that are relatively farfrom the origin indicate that such solutions are less likely to changein the near future. Therefore, well-conditioned solutions with verylarge eigenvalues, in terms of their distances from the origin, are lesslikely to undergo a change in matrix rank, which can be a criticalchange point, when the system parameters are undergoing slow drift. Achange in matrix rank occurs when one of the eigenvalues of a matrixwith temporally dependent entries approaches zero.

Furthermore, in one embodiment, the condition number may be multipliedby a constant factor. Alternatively, different types of matrix norms maybe used in the definition of the condition number, as can be appreciatedby those of ordinary skill within the art.

Referring back to FIG. 4, after the set of solutions is determined, thecontroller or other component may adjust the physical system beingmonitored and controlled based on the solutions determined (404). Forinstance, in relation to FIG. 3, the controller component 304 mayprovide feedback to the component 302 that is being monitored orcontrolled to adjust the component 302, or have the component 302adjusted. The controller further acquires updated parameters regardingthe physical system (406). For instance, in relation to FIG. 3, thecontroller component 304 may then monitor the component 302 to determinewhether any parameters have changed.

In one embodiment, the system parameters are measured at equally spacedtime intervals Dt. Alternatively, the time intervals used for monitoringthe system parameters may be unevenly spaced. As can be appreciated bythose of ordinary skill within the art, there are many different ways tointerpolate changes in the solutions to understand the drift of theparameters. For purposes of description, it is presumed that the systemparameters are acquired at homogenously spaced time intervals, althoughnon-homogenously spaced time intervals can also be employed. Inparticular, after each time t_(x+1)=t_(x)+Dt, the system parameters areacquired.

Each time the system parameters are measured, a set of updated solutionsfor the output feedback pole placement problem is determined basedthereon (408). More particularly, each solution that was previouslydetermined is updated based on the new system parameters. FIG. 6 shows amethod 600 that can be used for each solution when new system parametersare measured, according to an embodiment of the invention, to accomplish408 of the method 400 of FIG. 4. That is, the method 600 is performedfor each solution that was previously determined, when the systemparameters are updated.

First, if the updated parameters have not changed relative to theprevious parameters for the system (602), then the solution in questionis retained (604) without having to update the solution. That is, theupdated solution is equal to the preexisting solution previouslydetermined. However, if the updated parameters have changed relative tothe previous parameters (602), then the method 600 determines an updatedsolution relative to the new parameters. This updated solution isdetermined as has been described in relation to FIG. 4, using the samestarting point as the corresponding existing solution in question, or astarting point that is close to the existing solution in question, alongthe existing solution's original convergence path. Thus, an updatedsolution to the classical output feedback pole placement problem isdetermined using the projective approach of [Yang, Orsi, and Moore], asenhanced as has been described in relation to FIG. 4.

There are two possible outcomes upon determining the updated solutionfor the existing solution in question. First, the updated solution mayconverge after iteration. In this instance, the existing solution inquestion is stored for archival purposes, and further it is recordedthat determination of an updated solution was successful (608).Furthermore, the existing solution in question is replaced with theupdated solution that has been determined. In this way, the existingsolution is updated in light of changing system parameters.

Second, however, the updated solution may not converge after iteration.In this instance, the existing solution in question is again stored forarchival purposes, and further it is recorded that determination of anupdated solution was unsuccessful (610). Furthermore, one of two otheracts or steps is performed. First, the existing solution in question maysimply be removed from the set of solutions for the classical outputfeedback pole placement problem (612), since this solution is no longerrelevant for the updated system parameters. Second, the existingsolution in question may be replaced with one or more completely newsolutions (614). These completely new solutions are generated usingdifferent starting points than the existing solution and the updatedsolution that did not converge. The different starting points may berandomly selected in one embodiment of the invention, and theircorresponding completely new solutions are determined using theprojective approach of [Yang, Orsi, and Moore], as enhanced as has beendescribed in relation to FIG. 4.

It is noted that whether a solution is removed or replaced with one ormore completely new solutions when the updated solution does notconverge after iteration may be selected by the system manager beforethe system begins running. Alternatively, each time an updated solutiondoes not converge after iteration, the system manager may be able todecide whether the existing solution in question should be removed orreplaced with one or more completely new solutions. Finally, it is notedthat each time an updated solution does not converge, in one embodiment,after a number of such failed solutions in a given period of time, or ifthe percentage of solutions that have failed in the given period of timeis greater than a predetermined threshold, a system manager is alerted.

Referring back to FIG. 4, it is noted that 408 is performed, such as hasbeen described in relation to the method 500 of FIG. 5, for eachsolution at each time interval in which the system parameters aremeasured. Not depicted in FIG. 4 is that this process ends when theuser, such as the system manager, stops the system, or until nosolutions, and thus no approximate solutions, to the almost-static,classical output feedback pole placement problems are active. Forexample, all of the existing solutions may have been removed from theset of solutions, and/or there may be no viable completely new solutionsto replace the existing solutions.

Furthermore, each time the set of solutions is updated in 408—that is,after each time interval in which the system parameters are measured orupdated—410 is performed. Thus, the method 400 acquires the updatedparameters in 406, determines the set of updated solutions in 408, andthen performs one or more actions in 410. This process is repeated, asindicated by the arrow 420, until the user stops the system, or until nosolutions remain active, as has been described in the previousparagraph. Therefore, what is described next are the various actionsthat can be performed in 410 after the updated set of solutions has beendetermined for a given time interval's updated measured systemparameters in 408.

First, the physical system may be adjusted based on the updatedsolutions (412). For instance, in relation to FIG. 3, the controllercomponent 304 may provide feedback to the component 302 that is beingmonitored or controlled to adjust the component 302, or have thecomponent 302 adjusted. Second, the system manager may be notified(414). The system manager may be notified in one embodiment only whencertain critical changes in the system parameters occur, or may benotified at each time interval during which updated changed parametershave been measured. Third, and related to notification of the systemmanager, is that changes in the system, based on the updated solutions,may be monitored (416). Also related is fourth, that possibly criticalchanges in the system are detected, based on the updated solutions(418). These actions are now described in sum, even though they havebeen divided into separate actions in the method 400 of FIG. 4.

For example, the updated solutions can be used to construct a chart oran equivalent for monitoring the system. FIG. 7 shows an examplerepresentative chart 700 that can be constructed and displayed as a partof monitoring the physical system and notifying the system manager,according to an embodiment of the invention. As indicated in the column702, at time t=0, the initial set of solutions determined include thesolution matrices A₁, B₁, and C₁. As indicated in the column 704, at thetime t=Dt, the solution matrix A₁ has been modified as the updatedsolution matrix A₁+DA₁₁, whereas the solution matrices B₁ and C₁ remainthe same. The column 706 indicates the delta or change in the solutionmatrices between the time t=0 and the time t=Dt. Thus, the change in thefirst solution matrix is DA₁₁, whereas there is no change in the othertwo solution matrices.

As indicated in the column 708, at the time t=2Dt, the solution matrixA₁+DA₁₁ remains the same. The solution matrix B₁, however, has beenreplaced with the completely new solution matrix B₂. The solution matrixC₁ has been modified as the updated solution matrix C₁+DC₁₂. The column710 indicates the delta or change in the solution matrices between thetime t=Dt and the time t=2Dt. There is no change in the first solutionmatrix A₁+DA₁₁, while the replacement of the solution matrix B₁ with thecompletely new solution matrix B₂ renders this delta or changeinapplicable. Finally, the third solution matrix has changed by DC₁₂.

As indicated in the column 712, at the time t=3Dt, the solution matrixA₁+DA₁₁ has been modified as the updated solution matrix A₁+DA₁₁+DA₁₃.The solution matrix C₁+DC₁₂ and the solution matrix B₂ have remained thesame. The column 714 indicates the delta or change in the solutionmatrices between the time t=2Dt and the time t=3Dt. There is a change tothe first solution matrix by DA₁₃, whereas the second and third solutionmatrices have not changed.

As indicated in the column 716, at the time t=4Dt, the solution matrixA₁+DA₁₁+DA₁₃ has not changed, and the solution matrix C₁+DC₁₂ has notchanged. However, the solution matrix B₂ has been modified as theupdated solution matrix B₂+DB₂₄. The column 718 indicates the delta orchange in the solution matrices between the time t=3Dt and the timet=4Dt. There is a change only to the third solution matrix, by DB₂₄,such that the first and the second solution matrices have not changed.

Thus, information on each of the solutions may include parameters thatare inexpensive to determine. These parameters can include changes inthe largest eigenvalues, changes in the smallest eigenvalues, changes inthe matrix condition numbers, changes in the matrix Frobenius norm, andchanges in the slopes of the graphs of these changes, such as thediscretized accelerations of the parameters. Furthermore, in oneembodiment, the number of solutions that become obsolete during eachtime interval as well as over extended time intervals is tracked anddisplayed to the system manager as desired.

Changes in the number of solutions that become absolute as well as thediscretized acceleration of the number of solutions that become obsoletecan be used to identify possibly suspicious or unusually high numbers ofsolution failures. Such unusually high failures may indicate that asignificant shift in the system has taken place, such that an alert maybe automatically sent to the system manager. In addition, a gradual orconstant acceleration of the number of failures over more than a shortperiod of time may be an indication of system parts fatigue, so that analert may also be automatically sent to the system manager in thissituation.

The method 400 has been described in relation to solving a classicaloutput feedback pole placement problem. Furthermore, the method 400 canalso be extended to solve the generalized output feedback pole placementproblem. For instance, eigenvalues within the interior of closed regionsmay be selected, such as random, when generating the initial startingpoint matrices. Furthermore, if any of the regions is very close tozero, then the eigenvalues that are as large as possible and away fromthe origin are selected (and either as close to possible to the sizes ofthe other eigenvalues selected, or to one).

Thus, it is noted that, although specific embodiments have beenillustrated and described herein, it will be appreciated by those ofordinary skill in the art that any arrangement calculated to achieve thesame purpose may be substituted for the specific embodiments shown. Thisapplication is intended to cover any adaptations or variations ofembodiments of the present invention. Therefore, it is manifestlyintended that this invention be limited only by the claims andequivalents thereof.

1. A method comprising: determining a set of solutions for an outputfeedback pole placement problem based on parameters of a physicalsystem, the solutions being stable and well-conditioned to permitchanges to the parameters of the physical system to be monitored,wherein determining the set of solutions comprises determining the setof solutions for a generalized output feedback pole placement problembased on the parameters of the physical system by: a) selecting astarting point for determining a solution based on the parameters of thephysical system; b) determining the solution based on the startingpoint, assuming that the parameters are constant, based on a projectiveapproach; c) storing for the solution, a solution matrix, eigenvalues ofthe solution matrix, one or more final matrices of a solution path ofiterations leading to the solution, and eigenvalues of each final matrixof the solution path of the iterations leading to the solution; and, d)repeating a), b), and c) with different starting points for a number oftimes, such that the set of solutions are distinct; e) verifying thestability of each solution by: determining a condition number for thesolution matrix for the solution; determining distances of one or moresmallest of the eigenvalues of the solution matrix from an origin; f)selecting the set of solutions and matrices of the solutions that arewell-conditioned; adjusting the physical system based on the solutionsdetermined; acquiring updated parameters of the physical system at aplurality of time intervals; determining a set of updated solutions forthe output feedback pole placement problem based on the updatedparameters of the physical system by, for each solution, and based on aprojective approach: determining whether the updated parameters of thephysical system have changed relative to the parameters of the physicalsystem; where the updated parameters remain unchanged, determining theupdated solution as equal to the solution; and, where the updatedparameters have changed, determining the updated solution based on oneor more final matrices of a solution path of iterations that led to thesolution; where the updated solution converges after iteration,archivally storing the solution, indicating that determination of theupdated solution was successful, and updating the solution with theupdated solution; and, where the updated solution does not convergeafter iteration, archivally storing the solution; indicating thatdetermination of the updated solution was unsuccessful; removing thesolution from the set of solutions such that the solution is notutilized; generating at least one new solution; adjusting the physicalsystem based on the updated solutions determined; monitoring changes tothe parameters of the physical system based on the updated solutionsdetermined; and, detecting potentially critical changes in the physicalsystem based on the updated solutions determined.